Part 3

Tuning of PID Controls of Different Structures A. KAVA, The University of Akron, Akron, 0. and T. J. SCHEIB, Bailey Controls Co., Wickliffe, 0.

Although readily available as products, PID

controllers require tuning to suit a specific process. This article presents new data for optimized tuning of three PJD variations.

Proportional-integral-derivative ( PID) control is the most used controller type in industry and

is available as a stock item. However, its use is so diversified that the con­trol engineer must tune the PID values according to specific needs.

Studies have guided industry by providing quantitative data for tuning PID controllers for the given process, operational conditions, and perform­ance criteria. Unfortunately, these studies considered only one kind of functional PID control structure-the most popular one. Control hardware suppliers, on the other hand, have marketed other PID variations to bet­ter meet the requirements of a wide variety of control loops.

Recently, it has become easier to build PID variations due to low-cost microprocessor chip technology. But, the user does not always have suffi­cient guidance on how to tune these new PID controls. Some are probably using the tuning guides published for one form of PID, while some may not even know the PID form they have pur­chased. The objective of this article is to provide quantitative data as a tun­ing guide for three PID variations and to provide the user with an awareness of related industry practices.

Tuning factors

back control loop. The process and PID controller are shown in Figure 1, while some popular performance cri­teria are given in equations ( 5)- ( 7) .

In the published literature, the pro­cess, which is usually of high order, is approximated as a first order process

with a time delay, having a transfer function as in equation ( 1 ) :

G( s) Ke-os ---TS + 1

where, K is the process gain; r is the process time constant, sec; 8 is the process dead time (or delay), sec; and s is the Laplace domain variable.

Such approximation is made from a typical process reaction curve, as presented in Figure 2 (Reference 1). It is clear that the maximum slope of

There are three major factors in tuning a PID controller: the process, the con­troller's form, and the performance criterion (or response) of the feed-

FIG. 2: The process reaction curve, a response to change of controller output, m, is often ap­proximated as a first-order process. Delay time, 8, is found from the curve's maximum slope.

the curve determines the time delay magnitude, 0. Assuming first-order, the process reaction is (equation 2):

C(t) = Css ( 1 ~ e- tiT )

Here, for t = r, C( r) = 0.632C55,

which defines the r value from Figure 2. Note that c •• is the steady-state value of the process. Furthermore, the K value is found as K = ( c •• l u).

The controller considered in the published literature is an ideal PID control device that has an input! out­put relation and a transfer function re­lation as given by the following equa­tions ( 3 and 4):

where, m is the controller output; KG is proportional gain; T; is integral time, sec; and Td is derivative time, sec.

Popular performance criteria Next, three widely used controller performance criteria are introduced. Each is based on a different form of minimizing the error integral; namely: the minimum integral of square error ( ISE), the minimum integral of abso­lute error ( IAE), and the minimum in­tegral of absolute error multiplied by time ( ITAE) -as stated respectively in equations (5), (6), and (7).

00

ISE = f 0

e 2 dt

00

IAE = fo lei dt

00

ITAE = fo tiel dt

In the above equations, the error is

e = r~ c (refer to Figure 1). A graphical description of error, e, with respect to time, in Figure 3, helps to more clearly explain the error criteria.

There are other performance crite­ria used in the process industry. Ziegler and Nichols aimed at an over­all desirable response of 14 decay ra­tio, or ( bl a) = 0.25 in Figure 3. Oth­ers used a specified closed loop response in terms of response time.

The error, e, (see Figure 1) occurs from the set point changes or the dis­turbances (load changes) within a process, as shown schematically in Figure 3. Thus, the PID control should be tuned according to which criterion is considered important.

Disturbance or set point tuning Usually the process is subject to dis­turbances. However, in many cases a secondary (or cascade) control loop is present within the primary control loop. As the primary loop controller responds to disturbances, it adjusts the set point of the cascade control loop. This supports the idea that con­troller tuning for either the set point change or for the disturbance change can be important.

The popular tuning criteria, as wide­ly used in industry, have been the ISE, IAE, and ITAE. For an ideal PID con­troller given by equation ( 3), PID tun­ing values of these criteria have been studied and reported about twenty years ago, in References 1 to 3.

In these publications, the user was provided the tuning constants a through f and equations ( 8) and ( 9), with which to find the PID control val­ues. Note, that for given process pa­rameter values of K, r, and (), the PID control values of KG, T;. and Td are found for either the set point or distur­bance type tuning. But, the validity of the published tuning constants is re­stricted to small and moderate time delays, or 0 < (Oir) ::::::1.

So, for disturbance tuning ( eqs. 8):

KKG = a ( ()I r) b

and for set point tuning ( eqs. 9):

KKG = a ( () 1 r ) b

riT;=c+d(Oir)

Tdl r = e ( ()I r ) 1

Now, with the development of mi­croprocessors, different PID control functions have come into existence besides the ideal PID control. Among these are control types named: clas­sical, noninteracting, and industrial­as described by the next three equa­tions ( 1 0 to 12) , respectively. Each control type has transfer function re­lations similar to equation ( 4).

Classical:

1 ) ( + Tds m = KG ( 1 + T-s ---c---:--::::--- ) e , + r.s

Non interacting:

m = ( K + ~1- ) e ~ ( Tds 1 ) c G T;s r.s +

Industrial:

In these equations, T. is a filter time constant, usually defined as,

r. = 0.1 rd

See Figure 1 for the definition of other significant terms.

There are other PID structures in use in industry whose characteristics are discussed in References 4 and 5. However, the user needs to know the

FIG. 3: Two diagrams show typical variations of process value versus time. Response to step change of the set point is illustrated at the

left; while response to a step change of disturbance is given at the right. The shaded areas represent a graphical description of error.

quantitative values of K0 , T;, and Td of PID controls for each case (i.e., the process, the PID function, and the per­formance criterion) to obtain an opti­mum operation.

The purpose of the work reported in this article was to analyze and find those optimum tuning values for the new PID control variations presented in equations ( 1 0) through ( 12), and to compare their performances.

Optimization procedure The processes on which the optimum tuning constants were developed had specification parameters of r = 30 and K = 1. Delay time was varied from 8 = 3 to 30 sec. Such processes and controllers were simu­lated by ACSL (Advanced Continu­ous System Language).

The optimization procedure was conducted for (81r) = 0.1,0.5,and 1.0. A search procedure in the maxi­mum gradient direction was chosen as the controller values were changed to find the minimum value of the Per­formance Index.

Control values for each ( 8 I r) ratio were fitted into the curves of equa­tions ( 8) and ( 9) to find the tuning constants a through f. In doing this, the minimum least squares calcula­tion method was utilized.

To validate the optimum procedure, the ideal PID control was first used to generate the tuning values and com­pare them with those published 20 years ago. Due to potential variation in steps, up to a ten percent differ­ence between the values of the Per­formance Index was considered small and such results were considered compatible. The procedure of the study was validated in this manner.

Next, the same procedure was used for the various new PID controls and their corresponding tuning con­stant values, a through f, were found. Then, the values of error criteria were compared, when the different PID con­trols were tuned by the published tun­ing values of the ideal controller and by those found in this study.

New, optimum tuning values The tuning constants, a-f, for the new PID controls are significantly different from those published for the ideal controllers. The tuning constants de­veloped in this work are given for each controller type, in Tables 1 to 3.

In order to substantiate the superi­ority of the newly-found optimum tun­ing constants over those previously published for the ideal controller, tests were conducted to evaluate the values of the error criteria. The tests were conducted by using the new

FIG. 4: The above graph is an error comparison example of the Classical controller tuned for: ISE error criterion, a step change of disturbance input, and a process with (8 IT) = 0.5.

controllers on a process with ( 8 I r) = 0.5. The new PID controllers were tuned using published data-for the ideal PID controller-and by using th'3 data found in this work; then, the corresponding values of error criteria were compared.

The comparison indicates that in most cases, the values of error crite­ria are significantly lower when tuned by the optimum values found in this study, as opposed to those published for ideal controllers. As an example, one graphical comparison is shown in Figure 4. Here, the classical control­ler was tuned for the ISE criterion and for an input that corresponds to a step change of disturbance (see Figure 3, right). Note that the process re­sponse is unstable when tuned by the published data.

Conclusions Some suppliers of commercial con­trollers disclose the control functions of PID controls (References 4 and 5). Although the functions are described by different formats, they can be re­lated to the forms of equations ( 4) and ( 1 0) to ( 12), as given earlier.

It is recommended that the new tun­ing constants presented here be

adopted for the PID controllers with new functional relations. Further­more, the analysis should be extend­ed to the case of longer time delays, where ( 8 I r) > 1. Since the pub­lished literature has dealt only with the case of 0 < ( 8 I r) :::; 1 , the com­parative analyses of this work are also restricted to the same interval.

It is recommended that the case of longer process time delays be studied further. As a corollary, additional guid­ance should be given to the users as to what PID control structures should be employed with processes having such long delay times. 0

REFERENCES 1. Lopez, A. M., et al., "Tuning Controllers with

Error-Integral Criteria," Instrumentation Technol­

ogy, November 1967, pp. 57-62.

2. Miller, J. A., et al., "A Comparison of Control­

ler Tuning Techniques." CONTROL ENGINEERING,

December 1967, pp. 72-75.

3. Rovira, A. A., et al., "Tuning Controllers for Set

Point Changes," Instruments and Control Sys­

tems, December 1969, pp. 67-69.

4. Gerry, J.P., "A Comparison of PID Control Al­

gorithms," CONTROL ENGINEERING, March 1987,

pp. 102-105.

5. "Function Code Application Manual," 1£93-

900-20, Bailey Controls Co., Wickliffe, 0., 1986.

Pneumatic Instruments Gave Birth to Automatic Control MICHAEL BABB, CONTROL ENGINEERING

When engineers discovered that big valves could be delicately positioned with little puffs of air, they were on their way to inventing automatic control.

For most Americans, companies such as Boeing Aircraft, Union Carbide, and Dow Chemical are

household words . During the heady days of WW II manufacturing, they ral­lied their work forces to new pinnacles of productivity, and made a contribu­tion to the war effort that has received

high recognition ever since. However, it is unlikely that many

Americans have ever heard of compa­nies like Brown, Taylor, Foxboro, or Leeds & Northrup. Fewer still would recognize the contribution these be­hind-the-scenes instrument makers made, not just to WW II production,

but to the very way we carry on manu­facturing in the 1990s.

What exactly did engineers from these companies do? About 50 years ago, a handful of them, probably less than a few dozen, all told, invented what we now call automatic control.

The instruments they built-com­plex air-powered mechanical gadgets that mystified the layman and baffled even the operators who used them­paved the way for a new and revolu­tionary method for producing the ba­sic commodities of our industrial age.

Their new method readily adapted itself to liquids and gases, or anything , for that matter, that flows through a pipe. It was called continuous­flow processing.

The fruits of their labors made it possible for the allied forces in WW II to obtain tre­mendously large quantities of synthetic rubber and high-oc­tane aviation fuel. And tens of thousands of their mysterious instrument controllers, work­ing in concert at a single su­per-secret processing site, managed to squeeze out a few pounds of U-235 from gaseous uranium hexafluoride . Just enough, in fact, to begin the atomic age.

D

The heart, and lungs, of early propor­tional control is this force-balance mechanism. A temperature increase uncoils the Bourdon spring, moving the beam down to increase the nozzle baffle gap. The movement- a few thousandths of an inch at the most­decreases the nozzle back pressure, which through the relay causes an pro­portional increase in output pressure. At the same time, the increase in out­put pressure is fed back to the follow­up bellows, which restores the force beam to its original position.

The "Pre-Act" unit was Taylor 's in­novation. Its effect is to delay the feedback to the follow-up bellows so that the force beam does not come immediately back into equilibrium.

It all started with milk Pasteurization is not consid­ered a great engineering feat.

This gives the controller output an ex­tra "boost" whenever a change is made, a separate control action that

L----------------------1 became known as "derivative" control.

The temperature of a batch of milk is raised to 143 deg F and held there for 30 minutes. But pasteurization was, in fact, one of the first applications for process control. It was also one of the first processes that came under gov­ernment control.

Which probably explains why, about 1912, the dairy industry figured out that it had better start keeping records of the pasteurization of batches of milk. At that time, practically no one recorded temperatures, and so ther­mometer manufacturer Taylor lnstru-

ments, based in Rochester , N.Y. , didn't sell many of the mercury-re­cording thermometers they made.

The market changed in a hurry when Iowa-based dairy equipment manufacturer Cherry-Burrel put in an order with Taylor for 50 temperature

Conlrollu oulpul

(Above) Taylor Instruments ' technology breakthrough, the Fulscope controller, was the first PID

controller when it was introduced in 1940. At left, a battery of Fulscopes controls the sensitive temperature process of synthetic rubber making in Copolymer Corp.'s Baton Rouge, La., facility. A supervisor looks for perfect circles on the trend charts, an indication of good control.

recorders. A windfall was in the mak­ing. Taylor engineers and salesmen recognized the vertical opportunity and moved quickly to capitalize on it.

The original order in 1912 was the beginning of decades of domination of the dairy industry by Taylor. From their factory in Rochester came all sorts of customized temperature recorders, housed in ruggedized packages to withstand the daily cleaning-in-place. Taylor's temperature recorders be­came the de facto standard of the dairy industry. Competitors Foxboro and Brown were virtually locked out. Even as late as 1940, dairy sales of $625,000 accounted for 17% of Tay­lor's industrial business, according to Fortune magazine.

With its success in the dairy busi­ness, Taylor learned an important les­son in controls marketing. Once an end-user company buys into a specif­ic instrument line, and the operators become familiar with them, it is unlike­ly that they will ever switch to another brand. It is, in fact, virtually impossible to uproot an established instrument vendor; better to move in fast and lock out the competition early in the game.

But Taylor didn't rest on its laurels. Pasteurization technology changed in the mid-1930s. A new, nearly-continu­ous "flash" process heated the milk to 160 deg F for 15 seconds. To ac­commodate the speeded-up require­ments placed on the operator, Taylor invented a quick-acting diverting valve to flush the milk back into the heating tank if the temperature were not pre­cisely maintained. The instrument was, in fact, directly and automatically controlling a part of the process.

Continuous-flow processing The dairy process had been greatly accelerated, but it still was a batch op-

(Above) Fischer & Porter did not introduce pneumatic instrumentation until the mid-1940s; shown here is their "Concept 45" line (ca. 1960). (Left) The Bailey Meter Co. com­bined "Pilotrol" (proportional) and "Standa­trol" (integral) units into one controller. Bai­ley was an early controls industry leader for combustion and boiler control applications.

eration, as were all other chemical and refining processes in the early part of this century.

Probably the first real move towards continuous-flow processing was tak­en around 1925 by Carbide & Carbon Chemicals. The company was experi­menting with fractionation of natural gas in their West Virginia plant, and found they could produce new syn­thetic chemicals, such as ethyl alhohol and ethylene glycol. However, storing natural gas is a more cumbersome af­fair than storing, say, crude oil, and so Carbide was on the lookout for ways to speed the natural gas through their crackers more expeditiously. Maybe they wouldn't have to store gas at all.

Taylor was hired to supply the re­corders and a few controllers. But the technology team didn't work out. For

A Brown Instruments advertisememt in 1944.

one thing, Carbide was too secretive about its processes, and Taylor engi­neers didn't get a good grasp on the situation, as they had done in the dairy industry. Also, there was a major tech­nical barrier: the gas pipes required bigger and more tightly packed valves. It was hard to open and shut them, and Taylor's pneumatic actuator didn't do a good job. By 1933, Taylor came up with a suitable valve positioner. But Carbide operators had become skilled in manual continuous-flow operation, and didn't want to hear about it.

Meanwhile, deep in the heart of Texas, things were booming. The pe­troleum refiners were buying up all the crude the oil fields could produce. By 1929, they began to look at continu­ous-flow as a means of increasing their competitive advantage. They built continuous-cracking furnaces to handle the job, but due to the in­creased speed of the operation, refin­ers had to lean on instrument control more heavily than ever before. They also had a great need for more accu­rate flow measurement.

Enter Foxboro For years, Foxboro had been making instrumentation for the oil fields. It was practically the only company with enough flowmeter technology to get the job done.

But now Foxboro had a good op­portunity to expand its operation into the refineries. It didn't take long for entire complexes to become com­pletely outfitted with Foxboro instru­ment rigs. Taylor didn't even make a flowmeter, so they had no opportunity to compete. Meanwhile, Foxboro's in­struments and Stabilog controllers, housed in circular cases, became the de facto standard of the oil, gas, and refining industries.

The business was so good that by 1933, Brown introduced new instru­ments and had a go at it. While Fox­boro dominated, there was enough business opportunity for Brown to also have a significant share.

While Brown and Foxboro went af­ter the booming refining business, Taylor opted to go down a different path. It chose instead to concentrate on developing and improving the Fuls­cope controller.

In the mid-1930s, Taylor designed the "double response" unit which sta­bilized control action by providing for automatic valve reset. It was an inde­pendent unit attached directly to the valve stem, but when Taylor rede­signed the Fulscope, it was modified and integrated into the controller housing. A later generation of control engineers would refer to the reset as

"intergal mode" control. It was in 1938 that Tay­

lor engineers were mon­keying around with their controller and came across something new. John Ziegler describes the process of discovery:

"Taylor around this time was working the viscose rayon industry, trying to control the rayon shredder which was one of the god­awfulest pieces of chemi­cal engineering equipment ever devised. The propor­tional Fulscope with the double response unit would not work. Someone in the research depart­ment was tinkering with Fulscopes and somehow had got a restriction in the feedback line to the cap­sule that made the follow­up in the bellows. He not­ed that this gave a strange 'kicking' action to the out­put. They tried this on the rayon shredders and it gave perfect control on the temperature. The ac­tion was dubbed 'Pre-Act' and was found to help the control in other difficult applications, like refinery tube stills.

"The Pre-Act was the first derivative control and was incorporated into the Model 56R. It worked great on juice units in the sugar industry, but not so great in other applications."

Taylor then set out to design the Model 100 Fulscope. It was to incor­porate the automatic reset action provided by the double response unit, plus the new Pre-Act unit.

Taylor engineers John Ziegler and Nathaniel Nichols went to work on the problem. By 1941 they had a relatively straightfor­ward method for tuning the three-mode, or PID, controller. Basically, the method involves increas­ing the sensitivity (propor­tional response) until a sustained oscillation is ob­tained. Ziegler and Nich­ols called this the "ulti­mate sensitivity." Then, with the proportional ad­justment set to one-half the value that caused the ultimate sensitivity, set the reset (integral) rate equal to the ultimate frequency and set the Pre-Act (deriv­ative) to 1/8 of this fre­quency. This basic tuning formula, with variations, became known as the "Ziegler-Nichols" meth­od. It is a classic proce­dure that appears in every textbook on control engi­neering and continues to be used for PID controllers in our day.

Milk to atom bombs WW II came and the in­strument panic was on. Fortune magazine report­ed, "Synthetic rubber struggled through 1941 on a 1 0,000-ton-a-year bud­get, then doubled and re­doubled until drafting boards sank below the waves of blueprints for fif­ty-odd units to cost $750 million and produce over a million tons of synthetic rubber a year."

When the redesigned Fulscope went into ser­vice in 1940, it was a tech­nology breakthrough, the first three-mode controller on the market.

Foxboro's circular instruments and Stabilog controllers were common­place in the Texas oil and gas fields during the 1920s, and migrated into refinery control rooms and 1930s (photo: Premier Oil Refining Co.).

Carbide took on the un­likely butadiene-from-al­cohol assignment, the key ingredient of synthetic rubber. It boils only 1.9 deg C from its nearest chemical neighbor, bu-

There was a great deal of clever me­chanical engineering that went into the Fulscope. Special needle valves were designed for setting the reset and Pre-Act rates. A distinctive paral­lelelogram linkage was configured for continuous adjustment of sensitivity, or "gain" as it was later known. Al­though not advertised as such, Taylor made sure their instrument was oper­ator-friendly. Simple knob controls were all that was needed to adjust the

three modes, another industry first. By contrast, controllers from Brown and Foxboro were relatively difficult to tune and repair.

With its new controller, Taylor broke into Foxboro's Texas turf, notably at Texaco and Humble Oil refineries.

The Fulscope had one major draw­back, however. Setting the tuning pa­rameters for a two-mode controller had been difficult, but with the third mode, tuning became a major obstacle.

tene-1. This required very tight temperature control to draw them apart. "Companies that had never let a Taylor or Foxboro or Brown instru­ment through the gates," reported Fortune, "now scrambled their neat panel boards with a hodgepodge of instruments. The complexity and ve­locity of the new processes had out­run the ability of human operators to control them."

Similarly, to provide fuel for Roose­velt's 100,000 aircraft per year pro-

gram, the government poured $1 bil­lion in petroleum refineries. Fortune reported that production increased from 30,000 barrels a day in 1940 to 580,000 a day by 1945. "Instruments for the high-octane and synthetic rub­ber programs combined ran to a total of perhaps $40 million," reported For­tune. "Foxboro got the biggest share of this business, probably as much as both Brown and Taylor combined, with Taylor on the short end at about 15%, or $6 million."

But the Fulscope's great day was yet to come. In May, 1943, key Taylor executives met with representatives from the Manhattan Project in the Woolworth Building in New York City, who explained what they needed: a method for controlling the flow of gas­eous uranium hexafluoride as it passed through 4,000 barriers, with pumps and instrument control at eve­ry stage. Some of the instruments would have to measure the ''violently corrosive" gas in ranges of 0 to 0.125 psi; the best at that time measured 0 to 5 psi. It was the "trickiest flow job in history." And Foxboro, the acknowl­edged leader in flow control, had been passed over, probably because it was too busy producing instruments for other industries.

The job was to be done in Oak Ridge, Tenn. Kellex was contracted to built an 11-mile long gaseous diffusion plant along the Clinch River. The plant, named K-25, would require a staggering total of 200,000 instru­ments, of which 40,000 were actual controllers. The numbers amounted to about 25% of all the instruments made in the U.S. during WW II.

Taylor's response was to invent a new basic instrument, the pressure transmitter, and to produce them in prodigious quantities. Kellex ordered 30,000 of them, in 27 ranges. Taylor developed another twenty specials, and Kellex bought 70,000 of them.

When finished, K-25 had no less than 11 miles of instrument panels. So automated was the operation that Carbide, who ran the plant for Oak Ridge, bragged "only twenty opera­tors per mile are needed ... a figure that may get whittled down to ten." The end result was a few pounds of U-235, enough to build one bomb.

The postwar era Taylor's monumental efforts to supply instrumentation for the Manhattan Project doubled its industrial sales volume to $8 million in the first peace­time year, and for the first time topped Brown and Foxboro. But there were new worlds to conquer.

Leeds & Northrup had been the

Moore Products created a stir in 1946 with this "miniature" 5-in. Nul/malic controller.

leader in measuring temperature elec­trically. In 1941, Brown (purchased by Minneapolis-Honeywell in 1936) intro­duced its line of thermocouple-based "EiectroniK" temperature recorders and controllers. It caught Foxboro flat­footed and overturned the controls business in the refineries. At war's end, Foxboro rushed to develop its own line of electronic controllers, and the battle was on.

In the meantime, one of Taylor's

wartime subcontractors, Moore Products Co., moved into the pneu­matic foreground. Coleman B. Moore had left Brown Instrument in 1939 to form his own company, and by 1946, produced his first control product of fame: the Nullmatic "stack" control­ler. Measuring only 5 inches on a side, the Nullmatic eschewed the familiar circular pen-and-chart recorder; con­trol engineers, C. B. Moore figured, had learned to trust their instruments, and didn't require all the record keep­ing. The Nullmatic allowed construc­tion of dense control panels.

C. B. Moore went on to become a

One of the last great pneumatic control mno­vations was in 1965: Moore's Syncro Station.

leading exponent of pneumatic con­trols. Electronic controllers started gaining ground in the 1950s, and their safety had him worried. To demon­strate the inherent danger of electron­ics, and to amuse ISA audiences, he would dip his hand or a dollar bill into alcohol and set it afire with an electric current.

The one final significant achieve­ment in pneumatic controls was the introduction, by Moore Products in 1965, of the Syncro Station. It was­and still is, for many of them are still in use-a self-synchronizing controller which allowed simple, bumpless transfer from automatic to manual control. The Syncro has a "pneumatic memory" which is essentially a me­chanical memory built with a fluidic device with an air turbine driven lead screw.

Today, the great era of pneumatic controls has gone by. The electronic controllers that displaced them have themselves been largely replaced by digital electronics.

But, even though research and de­velopment had dwindled to nothing, pneumatic devices are still with us, and in much larger quantities than most would suspect. Their inherent safety is still unquestioned, and their durability virtually unchallenged. Many devices operating today are literally decades old, and going strong. 0

PID Controller Tuning Using Standard Form Optimization MICHAEL J. GILBERT POLONYI, Stridiron Services, Maspeth, N.Y.

A set of algebraic equations has been developed for dynamically exact controller settings based on loop information and the choice of a preferred transient response.

Tuning a process controller can be a frustrating experience be­cause of not knowing if and

when the setting was adequate-and just turning away, hoping that the complaints would turn away also. This article proposes some guidelines that may help in the task.

PID-controller tuning has remained an obscure art rather than an exact science. The reasons for this are sim­ple: nobody really knows what the set­tings should be since all criteria are qualitative in nature, and, due to the large self-regulatory capacity of most process systems, the margin of toler­ance is high and accurate settings are not necessary.

A fast response is always desirable. In practice, however, you may not get good results due to the unknown na­ture of many parameters-the system may become unstable, or stop con­trolling altogether. A slow response may then be necessary. (Another good reason to choose a slow re­sponse is that it is more efficient in energy terms.)

A variety of transient responses to step changes are possible, and a number of the responses that are con­sidered standard are described in the reference. Choosing a response from these standard forms for a specific behavior amount to optimization and is known as Standard Form Optimiza­tion, or SFO. By using SFO, a set of algebraic equations can be derived for each PID-controller. A few simple ex­amples of settings obtained by this method are shown in the table on pg. 1 06. These equations can also be used for the design of a self-tuning

controller by incorporating a time con­stant identification algorithm into the hardware of the controller.

System time constants The SFO method recommended here requires at least approximate knowl­edge of the system's three largest time constants. This is not a big prob­lem, once the basic understanding of

determining a time constant is mas­tered. Time constant determination has remained a rather esoteric con­cept in process control, and its useful­ness is not always duly recognized. There are three basic time constants: first-order, integration, and dead-time.

First-order and integration time con­stants-A first-order time constant T is 0.632 of the final value when a first­order system lag responds to a step input. A first order system is the same as an integrator on a feedback loop. Therefore, the integration time con­stant and a first-order time constant have the same value. Actually, an inte­grator is an idealization of a first-order system, because real integrators can exist only within a certain range, or else the output would grow unbound.

Even if it were physically possible for the output to grow unbound with­out saturating, the system would eventually force the integrator to wind down into a first-order type of re­sponse. Take for example the case of the integration time constant of a level control application-if the level were to increase unbound, eventually it would stop because the system pump pressure would equal the head of the liquid column. In other words, at some point the system will start experienc­ing some feedback of its own.

Dead-time time constant-Dead time is usually introduced into the control loop by the inability to measure the development of a process variable as it occurs. For example, to measure the effect on the temperature of a fluid coming from a heat exchanger, the temperature may have to be measured at a point downstream of the process to which the fluid is going, and it may take the fluid considerable time to reach that point. Another case is where chlorine is mixed in an open channel to kill bacteria. The residual chlorine sensor can be installed in the

channel only at a point where the pro­cess has already been completed, and this unavoidably creates a consider­able dead-time lag. Obviously, dead­time will also change if the velocity of the fluid changes, i.e., at different loads. This introduces an additional complication in the model, and the controller will have to be adaptive if this effect must be accounted for.

Dead-time is fairly simple to meas­ure in a control loop, because even when it can happen in any part of the loop, it will simply add up, and exactly where it originates is of no concern. In other words, there is only one dead­time constant per loop.

If dead-time is significant, i.e., one of the three largest time constants, it must be accounted for by using ap­propriate equations for tuning control­lers. It is not that dead time is hard to handle (especially with present con­troller technology) but it certainly af­fects adversely the speed of the re­sp'onse and, therefore, the quality of the control loop.

Dead time can be observed and measured by manually introducing a change on an otherwise stable opera­tion, and by measuring how long it takes to observe a response.

Recognizing a time constant Think of it this way-anywhere there is a time delay, there is a time con­stant. For example, heat exchangers, boiler furnaces, and nuclear reactors typically have first-order time con­stants between 1 0 to 20 sec.

Level control applications also have typical 10- to 20-sec time constants, but they can be easily changed since they depend on a very arbitrary choice-al-lowed level change.

Pressure time constants are usually much larger. For steam boilers, evap­orator main time constants range from 50 sec for once-through (supercriti­cal), 200 to 400 sec for water-tube models, and 30 to 45 min for fire-tube models.

Moments of inertia (relative to their maximum load for large rotating ma­chinery such as turbogenerators, pumps, and electric motors) are also integration time constants, and can range typically between 5 and 20 sec. The heavier the shaft respective to the maximum load, the higher the relative moment of inertia.

With the exception of the moment of inertia, electric motors and genera­tors have several electric time con­stants that can be ignored when deal­ing with electromechanical variations. They usually become significant when trying to control voltage and reactive power flow.

A first-order system (block diagram at top) is an integrator with a unit feedback loop. The first-order time constant T is 63.2% of the final value, and varies from the second-order.

When controller settings and re­sponse of the system are of major concern, main time constants can be determined through some relatively simple tests if they are large enough, i.e., more than 5 sec. For smaller val­ues, electronic measurements and in­struments are necessary.

Process equipment designers should be the first source to inquire when trying to determine time con­stant values, but usually they are themselves ignorant of this informa­tion. In any case, dynamic models and time constant information must be an­alyzed, described how it was ob­tained, and under what operating cir­cumstances that it applies, before approving the equipment. Otherwise, the whole concept may be flawed.

Any real system (mechanical, elec­trical or chemical) has many time con­stants. Fortunately, we are interested only in the largest ones. The higher the time constants, the slower the system will respond, which is good for the control engineer because there is plenty of time to take control action.

From a process point of view, this may be a limiting factor since fast load changes will not be responded to; i.e., the faster the natural response of the system the more accurate the control­ler action, and the settings, must be.

The natural response of a boiler, re­actor, and so forth, depends on its physical design characteristics. For example, a water-tube boiler will have a smaller main time constant than a fire-tube boiler, because the water content of the first is smaller than the second, for the same steam produc­tion, i.e., load.

Primary design rule A good first rule is to design with high time constants for equipment in processes that require slow changing loads. This is especially true if there are sudden bursts (of load) that aver­age out after a while. Also, design with low time constants if the process needs to respond primarily to load ramps.

For example, to control an exother­mic reaction in a batch process, tem­perature increases should be as slow as necessary in order to allow a stable heat dissipation. This will depend on the chemical reaction itself and the volume of the reactor. A large volume may cause hot spots and product burning. Therefore, the reaction itself can be slowed down by adding the re­agents slowly enough in the large-vol­ume reactor. The exothermic reactor time constants, in this case, will be: • The time it takes for the temperature

to reach its lower limit when addition of reactant has stopped and cooling is fully open;

• A rising temperature time constant when cooling is off and reactant valve is fully open. It must be larger (i.e., slower) than above in order to have a safe control setup. Note that a smaller time constant

means here to have a larger cooling than heating capacity-absolutely necessary to avoid a runaway tem­perature condition. Only the smaller cooling time constant counts for the adjustment of the controller settings. The ratio between the time constants above will determine the temperature swings during the reaction process. If temperature must be kept within small boundaries, a much larger cooling ca­pacity will be required.

Transmitter time constants Measurement is a crucial link in con­trol. If data coming from a transmitter are not reliable, they are useless. Data distortion can occur due to various reasons as follows: • Poor mechanical installation;

• Environmental specifications are exceeded;

• Poor choice of instrument; • Poor calibration stability (drift); • Too much relative instrument error; • Poor physical location.

Temperature sensors: Besides the measurment dead time due to installa­tion limitations as described previous­ly, temperature sensing elements can have significant first-order time lags (in the order of 50 sec) especially if they are installed in thermowells which introduce additional time lags. A bare thermocouple will give the fast­est-practically instantaneous-re­sponse, while a filled-system bulb in a well will give probably the slowest.

Flow sensors: Although flow meas­urement is always technically prob­lematic, the time lags (those associat­ed with the instruments and installation) are small, especially

where flow of liquids is involved.

Pressure sensors: Although pressure measurement is basically simple­and pressure transmitters are rugged, accurate, fast-some of the largest system first-order time constants are due to pressure variations as in high pressure steam boiler load control.

Level sensors: Level sensors can be finicky-especially if they are of the differential pressure type which need to be compensated for product densi­ty. Their installtion requirements and specification are very important. It is too common to specify the wrong sensor for the application. This will re­sult in erroneous readings which will translate to the rest of the loop.

Analyzer-type sensors: Analyzers (pH, 0 2, redox, dissolved oxygen, and photometric types) have, in general, become much more rugged. Analyz-

The top illustration shows the standard form of responses of Binomial (slow) and ITAE (fast) to a step input. The settings shown in the tables are those needed to obtain the standardized responses as shown by the curves. In the lower illustration, ITAE (fast) form of responses are shown. When the larger time constant is downstream of the controller, the servo and regulator responses are the same, but a farge overshoot of controller output occurs. A filtering and smoothing effect takes place when the largest time constant is located upstream of the control­ler, beneficially limiting controller output and the associated control effort of such a control loop.

ing transmitters are routinely being in­cluded in control loops, but they still require substantial maintenance, and redundant systems are necessary if they are to be heavily relied upon. In­stallation requirements will unavoid­ably introduce some dead-time lag.

Electromechanical sensors: Sensors for rpm, displacement, position, and so forth are, in general, very reliable and do not need calibration-and this is a major advantage. Accuracy and speed of response are major issues for all mechanical measurements.

Electric power system transducers: These sensors convert ac variables such as voltage, current, active and reactive power, frequency, and power factor, into linearized de rnA signals, or electric impulses. Most types need periodic test bench calibration if accu­racy and stability are of major concern which is often the case in frequency and active power measurements.

Instrument calibration Transmitters not used for indication, but for process control purposes only, need to be sensitive rather than accu­rate. This can make calibration a con­venient secondary priority if there are no indicating instruments connected to that same signal. Calibration-free instruments should be given a prefer­ence above all other types when avail­able. This usually means a pulse or digital output. The resolution will be one pulse, and accuracy will be the inverse of maximum pulse output.

In most process control applica­tions, a gain factor other than one is generated by the span of the transmit­ter and, to a lesser extent, by the siz­ing of the control valve. Rarely is a gain factor generated by the process itself. The controller then introduces only an adjustment factor to reach the dynamically optimum loop gain.

Controller settings and gains The following relationships can be ap­plied to all settings:

Proportional band (%) = G1 OO a1n

Integral time (sec) = Reset (re~~atsfmin)

Derivative action (sec) = Rate (re~~ats{min) The physical data of the working

conditions are required for most processes, e.g., pressures, maximum flow, level transmitter range, in order to be able to determine the gain of the transmitters that are a part of the con­trol loop. The transmitter data have to belong to the particular instrument

that is connected to the control loop. This is very important if the proper controller settings are to be obtained.

Since the gain of the controller will be affected by the gain of the transmitter and valve, it is neces­sary to determine these before at­tempting to adjust the controller gain. For example, in the case of a pressure transmitter, if it is cali­brated for 1,500 psi to be the full output, and for the output to be zero at 600 psi, the transmitter span will be: 900 psi, (i.e., 1,500 psi - 600 psi). If the system pres­sure setpoint is 1,000 psi, the transmitter gain will be as follows:

Tr n mitt r in = Setpoint value a 5 e ga Transmitter range

= 1,000 = 11 900 .

Transmitter gain is a major con­troller adjusting factor, and can easily range between 0.1 and 10. Control valve gain should not af­fect loop gain so dramatically. This must be verified and ac­counted for if other than one. Controller range, transmitter sig­nal range, and valve signal range must all be the same; e.g., 4-20 mA, 3-15 psi. Some controllers must have their output range limit­ed as part of the adjustments; for example, three-element drum lev­el control.

Some valve actuators may have smaller ranges. This is the case for control valve split-range de­signs. But what counts is the combined gain of both valves working together as one from the controller.

If a change in signal occurs within a loop as, for example, with the use of an electropneumatic transducer, the ranges must all correspond on a 0 to 1 00% scale basis. In other words, full scale and zero scale correspondence must be maintained; otherwise a new gain coefficient is intro­duced. The equations presented here do not make any provisions for changes in signal range.

With the exception of dead time, linearity is assumed overall. That is, transmitters and control valves have inherently linear re­sponses. PID controllers are linear by definition.

Dynamic models Characterization is not accounted for. All transmitters are assumed to be lin­earized for any flow, pressure, or tem-

Servo and Regulator Loops

In any control loop, two outputs can be defined for each input, and they are called a Servo output and a Regulator output. They are futher defined as follows:

lim c(t) Servo: t--'>oo i(t) = 1

lim r(t) Regulator: t--'>oo d(t) = 0

with i(t) = d(t), since the input d(t) to a Regulator is also called a"disturbance."

The Laplacian forms for the servo and regulator in the block diagrams are:

C(s) Pure Servo: l(s)

R(s) Pure Regulator l(s)

(Piant)(PID)

1 + (Piant)(PID)

1

1 + (Piant)(PID)

perature condition, as are valves. The controller settings derived and

listed in the table on the next page are based on third-order dynamic models for PI controllers, and on fourth-order dynamic models for PID controllers. In this way, exact settings for gain, reset, and rate will result from the closed-

loop model by equating the char­acteristic equation to a fast re­sponse (ITAE) or slow response (binomial).

Determining a time constant re­quires more ingenuity than work. quires more ingenuity than work. The following items are some general guidelines that may prove to be useful.

Empirical method-Measure an input step response as follows: • Stabilize the process by either

transferring the load fluctuations to another unit, or by arranging a period or interval at constant load with the production people;

• Set the loop to be measured in manual mode;

• Introduce a step change in the system. This step must be either a load change (e.g., steam flow) or a control variable step (e.g., fuel);

• Measure the response of the af­fected variable. Instruments for this purpose must be accurate and reliable. If the response is slow enough, take readings at fixed time intervals and plot them. If the response is too fast, a plotter must be used;

• The time constant is the time it takes for the initial slope to inter­sect the final value (see re­sponse curves, opposite). All readings must be converted to a percentage of the physical vari­able, by defining a "base" value;

• This method requires some trial and error until a satisfactory set of readings has been obtained, but has the advantages of: not disrupting the operation too much; use of minimum amount of instruments; and usually are of short duration (approximately ten minutes for each test, with one or two minutes of actual tak­ing of readings);

• Determine the largest time con­stants of the system, and list them beginning with the largest one and down. Stop when the next is the controller, or if there are three time constants.

Analytical method-This method is recommended when all relevant parameters are known, which in

real life is rarely the case. There is one situation in which this

is especially useful-level control. A typical question is: What is the main time constant of a tank or drum? The answer is, the maximum allowed level fluctuation times the tank surface, di­vided by the maximum flow.

This is a case similar to a chemical reactor temperature time constant, since two time constants can be de­termined: one for rising level and one for dropping level. But only one is in the control loop and that is the one that counts for the controller settings.

The maximum allowed level fluctua­tion is not necessarily the transmitter range, but usually a much smaller val­ue since normal operating conditions are being considered. The level that the controller maintains before the lev­el alarm goes off must also be consid­ered. Therefore, a level transmitter usually introduces a gain factor much smaller than one. This transmitter gain must be compensated for by the controller by adjusting its own gain.

The largest time constants do not always come from the process or the plant. They also come from the control equipment itself; the control valve, for instance, or the pneumatic signal tub­ing, or the transmitter. This is very well the case when controlling flow with a flowmeter and control valve on the same pipeline. These cases re­quire that the time lag of the controller be much lower than the time lag of the transmitter, and the error of the trans­mitter much lower than the error of the valve. Otherwise, the purpose of the control loop is self-defeating. Al­though widely used, this kind of "self­control" loop should be avoided be­cause it slows down the response of the control system. Note that "much lower" or "much higher" means, in practical process control terms, a fac­tor of ten or more.

On-line system identification Virtually all process control loops can be separated into two parts: the plant and the controller. Since it is known what the controller can do, the major task is to model and identify the plant with the transmitters and control valves. With PID controllers, only three plant time constants can be handled. Therefore, a model is already speci­fied as follows:

Plant =Gain and the three largest time constants.

This makes the identification pro­cess substantially easier. By measur­ing at regular intervals the signals of the controller to the plant, and reading back the plant's response, the actual values can be determined. Of course, the plant's response will be contami­nated by noise coming from the pro­cess disturbance input, but this noise can be eliminated by a filtering algo­rithm which can easily be implement­ed in a digital controller.

Although some initial gain estima­tion by the old-fashioned paper and

pencil method, and obtaining dead time by stop-watch are easy, these also can be determined by completely automatic means. A primary rule to re­member is that process plant gain originates in the transmitters and the control valves. D

REFERENCES 1. D. Graham and R. C. Lathrop, The Synthesis of "Optimum" Transient Response: Criteria and

Standard Forms, Transactions of AlEE, Vol. 72, Part

II, November 1953, Pg 281.

2. M. Polonyi, "Boiler Dynamic Load Control and

Optimization," CONTROL ENGINEERING, Sept. 1985